Letwith values in a Euclidean space will be denoted by .
We consider the following optimization problem
and is a random element in some measurable space with an unknown probability law . The function is assumed continuously differentiable (for each ) but it can possibly be non-convex. Suppose that one has access to i.i.d samples drawn from , where is fixed. Our goal is to compute an approximate minimizer such that the population risk
is minimized, where the expectation is taken with respect to the training data and additional randomness generating .
Since the distribution of is unknown, we consider the empirical risk minimization problem
using the dataset
Stochastic gradient algorithms based on Langevin Monte Carlo have gained more attention in recent years. Two popular algorithms are Stochastic Gradient Langevin Dynamics (SGLD) and Stochastic Gradient Hamiltonian Monte Carlo (SGHMC). First, we summarize the use of SGLD in optimization, as presented in [RRT17]. Consider the overdamped Langevin stochastic differential equation
where is the standard Brownian motion in and is the inverse temperature parameter. Under suitable assumptions on , the SDE (3) admits the Gibbs measure as its unique invariant distribution. In addition, it is shown that for sufficiently big , the Gibbs distribution concentrates around global minimizers of . Therefore, one can use the value of from (3), (or from its discretized counterpart SGLD), as an approximate solution to the empirical risk problem, provided that is large and temperature is low.
In this paper, we consider the underdamped (second-order) Langevin diffusion
where, model the position and the momentum of a particle moving in a field of force with random force given by Gaussian noise. It is shown that under some suitable conditions for , the Markov process is ergodic and has a unique stationary distribution
where is the normalizing constant
It is easy to observe that the -marginal distribution of is the invariant distribution of (3). We consider the first order Euler discretization of (4), (5), also called Stochastic Gradient Hamiltonian Monte Carlo (SGHMC), given as follows
where is a step size parameter and
is a sequence of i.i.d standard Gaussian random vectors in. The initial condition may be random, but independent of .
In certain contexts, the full knowledge of the gradient is not available, however, using the dataset
, one can construct its unbiased estimates. In what follows, we adopt the general setting given by[RRT17]. Let be a measurable space, and such that for any ,
where is a random element in with probability law . Conditionally on , the SGHMC algorithm is defined by
where is a sequence of i.i.d. random elements in with law . We also assume from now on that are independent.
Our ultimate goal is to find approximate global minimizers to the problem (1). Let be the output of the algorithm (9),(10) after iterations, and be such that . The excess risk is decomposed as follows, see also [RRT17],
The remaining part of the present paper is about finding bounds for these errors. Section 2 summarizes technical conditions and the main results. Comparison of our contributions to previous studies is discussed in Section 3. Proofs are given in Section 4.
Notation and conventions. For , scalar product in is denoted by . We use to denote the Euclidean norm (where the dimension of the space may vary). denotes the Borel - field of . For any -valued random variable and for any , let us set . We denote by the set of with . The Wasserstein distance of order between two probability measures and on is defined by
where is the set of couplings of , see e.g. [Vil08]. For two -valued random variables and , we denote . We do not indicate in the notation and it may vary.
2 Asumptions and main results
The following conditions are required throughout the paper.
The function is continuously differentiable, takes non-negative values, and there are constants such that for any ,
There is such that, for each ,
There exist constants such that
For each , it holds that
There exists a constant such that for every ,
The law of the initial state satisfies
where is the Lyapunov function defined in (16) below.
If the set of global minimizers is bounded, we can always redefine the function to be quadratic outside a compact set containing the origin while maintaining its minimizers. Hence, Assumption 2.3 can be satisfied in practice. Assumption 2.4 means that the estimated gradient is also Lipschitz when using the same training dataset. For example, at each iteration of SGHMC, we may sample uniformly with replacement a random minibatch of size . Then we can choose where are i.i.d random variables having distribution . The gradient estimate is thus
An auxiliary continuous time processes is needed in the subsequent analysis. For a step size , denote by the scaled Brownian motion. Let be the solutions of
with initial condition where may be random but independent of .
There exists a constant such that for all ,
The proof of this theorem is given in Section 4.2. ∎
The following is the main result of the paper.
The proof of this theorem is given in Section 4.3. ∎
Let . We have
3 Related work and our contributions
Non-asymptotic convergence rate Langevin dynamics based algorithms for approximate sampling log-concave distributions are intensively studied in recent years. For example, overdamped Langevin dynamics are discussed in [WT11], [Dal17b], [DM16], [DK17], [DM17] and others. Recently, [BCM18] treats the case of non-i.i.d. data streams with a certain mixing property. Underdamped Langevin dynamics are examined in [CFG14], [Nea11], [CCBJ17], etc. Further analysis on HMC are discussed on [BBLG17], [Bet17]. Subsampling methods are applied to speed up HMC for large datasets, see [DQK17], [QKVT18].
The use of momentum to accelerate optimization methods are discussed intensively in literature, for example [AP16]. In particular, performance of SGHMC is experimentally proved better than SGLD in many applications, see [CDC15], [CFG14]. An important advantage of the underdamped SDE is that convergence to its stationary distribution is faster than that of the overdamped SDE in the -Wasserstein distance, as shown in [EGZ17].
Finding an approximate minimizer is similar to sampling distributions concentrate around the true minimizer. This well known connection give rise to the study of simulated annealing algorithms, see [Hwa80], [Gid85], [Haj85], [CHS87], [HKS89], [GM91], [GM93]. Recently, there are many studies further investigate this connection by means of non asymptotic convergence of Langevin based algorithms and in stochastic non-convex optimization and large-scale data analysis, [CCG16], [Dal17a].
Relaxing convexity is a more challenging issue. In [CCAY18], the problem of sampling from a target distribution where is L-smooth everywhere and -strongly convex outside a ball of finite radius. They provide upper bounds for the number of steps to be within a given precision level of the 1-Wasserstein distance between the HMC algorithm and the equilibrium distribution.
Our work continues these lines of research, the most similar setting to ours is the most recent paper [GGZ18]. We summarize our contributions below:
Diffusion approximation. In Lemma 10 of [GGZ18], the upper bound for the 2-Wasserstein distance between the SGHMC algorithm at step and underdamped SDE at time is (up to constants) given by
see Theorem 2.8. This upper bound is better not only in convergence rate for both step size ( vs. ) and variance ( vs. ) but also in the number of iterations. This improves Lemma 10 and hence Theorem 2 of [GGZ18]. Our analysis for variance of the algorithm is also different. The iteration does not accumulate mean squared errors, as the number of step goes to infinity.
Thanks to the big data regime, dependence structure of the dataset in the sampling mechanism, can be arbitrary, see the proof of Theorem 2.8. The i.i.d assumption on dataset is used only for the generalization error. We could also incorporate non-i.i.d data in our analysis, see Remark 4.5, but this is left for further research.
4.1 A contraction result
In this section, we recall a contraction result of [EGZ17]. First, it should be noticed that the constant and the function in their paper are and in the present paper, respectively. Here, the subscript stands for “contraction”. Using the upper bound of Lemma 5.1 for below, there exist constants small enough and such that
Therefore, Assumption 2.1 of [EGZ17] is satisfied, noting that and
We define the Lyapunov function
For any , we set
where are suitable positive constants to be fixed later and is continuous, non-decreasing concave function such that , is on for some constant with right-sided derivative and left-sided derivative and is constant on . For any two probability measures on , we define
Note that and are semimetrics but not necessarily metrics. A result from [EGZ17] is recalled below.
For a probability measure on , we denote by the law of when .
There exists a continuous non-decreasing concave function with such that for all probability measures on , we have
where the following relations hold:
The function is constant on , on with
and satisfies .
See Theorem 2.3 and Corollary 2.6 of [EGZ17]. ∎
It should be emphasized that , and consequently, contracts at the rate .
4.2 Proof of Theorem 2.8
For each , we define
Let be -valued random variables satisfying Assumption 2.6. For , we recursively define , and
Let . For each , and for each , we set
We estimate for ,
Denote . By Assumption 2.4, the estimation continues as follows
Using (22), one obtains
noting that Therefore, the estimation in (23) continues as
Applying the discrete-time version of Grönwall’s lemma and taking squares, noting also that yield
Taking conditional expectation with respect to , the estimation becomes
Since the random variables are independent, the sequence of random variables , are independent conditionally on , noting that is measurable with respect to . In addition, they have zero mean by the tower property of conditional expectation. By Assumption 2.4,
Doob’s inequality and (26) imply
Taking one more expectation and using Lemma 5.3 give